In mathematics, extendible cardinals are large cardinals introduced by Reinhardt (1974), who was partly motivated by reflection principles. A cardinal number κ is called η-extendible if for some λ there is a nontrivial elementary embedding j of
into
where κ is the critical point of j.
κ is called an extendible cardinal if it is η-extendible for every ordinal number η.
Vopenka's principle implies the existence of extendible cardinals. All extendible cardinals are supercompact cardinals.
"A cardinal κ is extendible if and only if for all α>κ there exists β and an elementary embedding from V(α) into V(β) with critical point κ." -- "Restrictions and Extensions" by Harvey M. Friedman http://www.math.ohio-state.edu/~friedman/pdf/ResExt021703.pdf